Visible crew/equipment: When Clara and Opie first sit down at the piano to begin the piano lesson, note there's nothing on the floor behind them. Then that shot cuts to the end of the lesson as the camera pulls back for a wideshot, and now there are two tape marks on the floor near the door in the area where Opie and Aunt Bee will be standing shortly. (00:04:15)
Visible crew/equipment: After the boys meet Flip Conroy and talk about the football team, when they all disperse and Opie starts walking away the T-mark is visible on the ground.
Visible crew/equipment: At the start, just as Mr. Hampton walks into the office and pushes the door closed, we can see a crew member's hand reaching through the doorway to help close the door. (00:00:55)
Visible crew/equipment: After the squad car is hit from behind Andy rushes over to Goober, and a T-mark (it's a bit darker than the color of the dirt road) becomes visible on the ground (just as the woman gets out of her car) at the bottom of the screen, right behind Andy's feet. (00:01:25)
Aunt Bee's Crowning Glory - S7-E5
Visible crew/equipment: When Andy removes the mirror from the wall (for Aunt Bee), the reflection of a studio light is visible in the mirror just as Andy starts moving it. (00:18:40)
Visible crew/equipment: When Aunt Bee and Opie leave the house and get in the car, the reflection of crew and filming equipment can be seen on the surface of the car.
Visible crew/equipment: Just as Aunt Bee and Opie pull up in front of Wally's station, the reflections of crew and equipment are visible on the surface of the car. (00:06:00)
Visible crew/equipment: When Flip pays a visit to the Taylors, he and Andy walk over to Opie, and the actors' two tape marks are visible on the floor between Opie and Flip. (00:21:05)
Visible crew/equipment: When Aunt Bee drives her car up to the gas pump at Wally's station and then wins the five dollars, the reflections of crew and filming equipment are visible on the surface of her car. Also, note that the chrome trim around the windshield is covered with black tape, presumably to prevent reflection. Additionally, during all the scenes at Wally's station, the back end of one of the quonset huts from the "Gomer Pyle, USMC" set are visible, either at the left side of the screen or in the background.
Aunt Bee's Restaurant - S7-E21
Visible crew/equipment: After Goober orders the chow mein dinner at the restaurant, in the following wideshot of everyone at the table, the curved chalk actor's mark is visible on the floor between Andy's and Helen's chairs. (00:12:20)
A Visit to Barney Fife - S7-E18
Visible crew/equipment: When Barney introduces Andy to Ma's boys, the actors' two tape marks are visible on the floor in front of the couch. (00:07:10)
Don't Miss a Good Bet - S7-E16
Visible crew/equipment: When Andy walks into the diner to speak with George Jones about investing in the treasure hunt, the shadow of the overhead boom arm and mic is moving around on the wall behind them. (00:18:30)
Visible crew/equipment: When all the Darlings walk out of Andy's kitchen with a bag full of Aunt Bee's muffins, the actor's tape mark is visible on the floor, where one of the Darling boys looks down and stands. (00:20:25)
Answer: It was a joke. They were supposed to be dumb hillbillies.
Both. According to later interviews with both Andy and the Dillards band (the real name of the band that played the parts of the Darling children, of which Charlene (Maggie Peterson - Mancuso was also a member) the reason the Darling boys never had any lines was because they would've had to have been paid if they spoke. In return for appearing without lines or pay, Andy promised to get as much of their music onto the show as possible which, he did. This obviously gave them a lot of exposure.